Differential manifold without connection: Is it possible?
We all are familiar with the kind of differential geometry where some affine connection always exists to relate various tangent spaces distributed over the manifold, and from this connection two fundamental tensors, namely the Cartan's torsion and the RiemannChristoffel curvature, arise.
Is it possible to have a differential manifold, where due to some topological anomaly, a connection cannot exist? Of course there exists symplectic manifolds where no connection property is required. But my question is related to the existence of nonmetricity  as in nonmetric case, metric property id there but the connection is not metric. Similarly, can we have some property called nonconnectivity where the affine connection is there but somehow it is lacking some fundamental connection requirements? 
Re: Differential manifold without connection: Is it possible?
Quote:

Re: Differential manifold without connection: Is it possible?
...hence, if by manifold you mean, as differential geometers do, a space which is paracompact (or something stronger like second countability), then every (smooth) manifold admits a Riemannian metric and a connection compatible with it.

Re: Differential manifold without connection: Is it possible?
Quote:

Re: Differential manifold without connection: Is it possible?
Quote:
I'm an engineering student. I asked these question regarding nonRiemannian description of defects in solids. In this field, there arises a manifold which is composed of disjoint noncompact parts and you cannot form a compact Euclidean subset from these parts by an unique global diffeomorphism. 
Re: Differential manifold without connection: Is it possible?
Quote:

Re: Differential manifold without connection: Is it possible?
Could you give more information about this manifold? If it's a manifold, every point has a compact neighborhood about it. The proof that every manifold has a connection is based on the assumption that you can form a partition of unity. If your manifold is not paracompact, this fails. However, most definitions of manifold assume this to begin with.

Re: Differential manifold without connection: Is it possible?
Quote:

Re: Differential manifold without connection: Is it possible?
Quote:
I am told that this is the first example ever discovered. It is a ten dimensional manifold. I do not understand the paper but would be willing to read it through with you. 
Re: Differential manifold without connection: Is it possible?
many thanks for the discussion.

All times are GMT 5. The time now is 02:25 PM. 
Powered by vBulletin Copyright ©2000  2014, Jelsoft Enterprises Ltd.
© 2014 Physics Forums